I guess by now most people have heard about the new paper (arXiv:1109.4897) by the OPERA collaboration which claims to have observed superluminal neutrinos with 6$\sigma$ significance. Obviously this has been greeted with a great deal of skepticism, and there will no doubt be debate over systematic errors for a long time to come (and frankly I expect some unaccounted for systematic error to be the case here).

Obviously theorists abhor superluminal travel, and I am well aware of many of the reasons for this.

However, the paper has me wondering whether there have been any toy models put forward which would be both consistent with the OPERA paper, and with earlier bounds on neutrino velocity.

In particular, if taken with other previous papers (from MINOS and from observations of the 1987 supernova) you have the following bounds on neutrino velocity in various average energy regimes:

Is there any proposed model which is actually consistent with such results? It seems that there has been a lot of pointing to the supernova bound (the 10MeV scale) as being inconsistent with the reported findings. However if there was a mechanism whereby the velocity were a monotonic function of energy (or depended on flavor), this argument would be negated. Do there exist any such proposed mechanisms?

Do you mean "if there was a mechanism whereby the velocity were a non-monotonic function of energy"?
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SlaviksSep 23 '11 at 6:39

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@Slaviks: No. I mean that the bounds are further away from $c$ for increasing energy, and this could be explained by a monotonic function, which seems more natural than a non-monotonic one. I guess you are refering to the 3GeV data point, but here the error bars are loose, so it could easily be below the 17GeV velocity while still being within the error bars. But, frankly I'd be interested in any models which fit all 4 points relatively well.
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Joe FitzsimonsSep 23 '11 at 6:43

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@Michael: What in particular? The superluminal bit? Yep, it's crazy, and everyone seems to be rightly skeptical, but who knows? It's unlikely we will know for sure until other experiments either replicate or fail to replicate the results. Even the OPERA team seems skeptical of their own results.
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Joe FitzsimonsSep 23 '11 at 8:35

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@MichaelKissner As far as I can see it doesn't fit into string theory. String theory also has c as the speed limit.
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Kelly DavisSep 23 '11 at 9:03

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@Kelly, at least this could give me a new opening for my Thesis :) "Ignoring [1], we..."
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MichaelSep 23 '11 at 9:50

4 Answers
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I am afraid that one has to go to a "very unusual segment" of theoretical literature if he wants any papers about superluminal neutrinos. Guang-jiong Ni has been authoring many papers about superluminal neutrinos a decade ago:

and probably others. They are pretty much cited by the same author only so you may become the second person in the world who has read them. For somewhat more well-known papers on tachyonic neutrinos, see

which were raised by the observations of apparently superluminal neutrinos in the decay of the tritium atoms. Well, the older ones were written before the tritium atom decay anomaly. An even older paper is

which reviewed the experimental situation of tachyonic neutrinos as of 1985. You may want to check many more papers by Alan Kostelecky because he's been working on similar possible ways how the Lorentz symmetry could be broken for decades and he is a rather serious researcher. See also

where two Weyl equations were joined into a twisted Dirac equation of a sort. Not sure whether it made any sense. On Sunday, I will post an article on my blog about a vague way to get different speeds of light from noncommutative geometry (in string theory or otherwise):

As you noted as well, the functional dependence of the velocity on the neutrino energy would have to be an extremely unusual function which de facto invalidates the Opera results without any loopholes. However, there could be a loophole: the neutrino could become highly tachyonic only while it moves through the rocks. "Index refraction for neutrinos" could be smaller than one for common materials such as rocks. It sounds of course as incompatible with relativity as the tachyonic neutrinos in the vacuum but by splitting the experimental data into the vacuum data and rocks data, you could get more sensible velocity dependence on energy in both cases.

Thanks Lubos for the detailed answer. For what it's worth, I've also just come across arXiv:0805.0253 by John Ellis and others which examines energy dependent velocities, and doesn't seem to be entirely incompatible with the current results. Thanks for a great answer, and welcome to the site.
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Joe FitzsimonsSep 23 '11 at 18:25

Right, @Joe, but is that correct to say that the Ellis et al. paper doesn't try to offer any theory, just a functional description of possible observations? Thanks for welcoming and for the very interesting question.
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Luboš MotlSep 23 '11 at 18:59

@Joe, I think Luboš is right -- that paper (note that there's some overlap of its author list and OPERA!) is more of an ansatz than a model. And as far as I can tell it blatantly disagrees with everything we know about effective field theory.
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Matt ReeceSep 23 '11 at 23:59

Yes, sorry, I should have been clearer on that. I just thought it was interesting in the context of the energy dependence loophole.
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Joe FitzsimonsSep 24 '11 at 3:43

@LubošMotl: Hi Lubos. Thanks for accepting my invitation. If you have any questions which you think I might be able to assist you with, don't hesitate to send me a message via fb / Google+. -- Regards
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UGPhysicsSep 24 '11 at 6:31

It's very hard to imagine that there is any sensible model consistent with OPERA's results. (Aside from models of unaccounted-for systematic uncertainties in the experiment.) We know that we live in a world described to very high precision by Lorentz-invariant quantum field theory, so the most sensible way to look for Lorentz violation is to start with such a theory and add small Lorentz-breaking terms to the Lagrangian. Because we assume that they're small, the usual dimensional analysis that tells you if operators are relevant, marginal, or irrelevant in effective field theory still applies. So, for instance, instead of a particle having a kinetic term $\eta^{\mu\nu} \partial_\mu \phi \partial_\nu \phi$, we can add a piece $\epsilon \partial_t \phi \partial_t \phi$ that picks out a preferred time direction. This continues to be dimension 4, so it's a marginal operator -- you should expect such operators to be present with order-1 coefficients, and they should run logarithmically just as in usual field theory, so if you try to set all of them to zero except the ones for neutrinos, say, the others will be generated nonetheless. This is already a bad sign for the viability of such theories.

The Lorentz-violating extension of the SM, defined along these lines, has been studied by Coleman and Glashow and by Colladay and Kostelecky. (As Lubos mentions, Kostelecky has done a lot of careful and serious work on understanding exactly how well Lorentz invariance has been tested.) It turns out that there are not just marginal operators, but also relevant ones, which makes the whole picture look even worse. However, the relevant operators all violate CPT, so if you decide to consider only theories that violate Lorentz invariance while preserving CPT, you're on slightly safer ground, although marginal operators are not by any means safe. The bounds are extremely strong: a 2008 review by Kostelecky and Russell has convenient tables of various operators. From there you can find references to other papers like this one of Altschul that derives a bound of about $10^{-11}$ on the analogue of "$\epsilon$" for muons. If you want a theory where muon neutrinos see a violation of the speed of light, you'll inevitably generate such a thing for muons -- even if you cleverly try to arrange it to depend on electroweak symmetry breaking to evade the fact that they live in the same doublet, loops will inevitably generate the term for muons, at a level larger than $10^{-11}$. Electrons are even more strongly constrained, at the $10^{-15}$ level, and because neutrinos oscillate into different flavors, you can't avoid confronting that bound as well.

That's not to mention the supernova 1987A constraints on neutrinos, which tell you that if you want all this to work you need to go to even more exotic theories and look at energy dependence, as you noted in the question. You can try to do that with some higher-dimension operators, but again, effective field theory tells you that you can never do that sort of thing in isolation. Break the symmetry somewhere, and all allowed terms are generated. And they're all very well bounded by data. (The size of the effect tells you that these higher-dimension operators would be suppressed by relatively low scales, which is also not encouraging.)

I won't claim that I've given you a completely airtight argument here, but anyone who really wants to claim to have an explanation of how a particle can go faster than $c$ has to confront these effective field theory issues. Anyone who claims to be able to just sidestep them completely is either trying to fool you, or fooling themselves.

Edited to add: Because I think Moshe is right that this point isn't widely enough appreciated, while I'm on my soapbox I might as well point out for readers not well-versed in effective field theory that precisely the same argument should be deployed against the idea that physics is fundamentally discrete, or that we might be living inside a condensed matter system that flows near a Lorentz-invariant fixed point, or all sorts of other new kinds of science that people might try to sell you.

Wish I could upvote this more than once, more people should be aware of this issue. But, potentially stupid question, does this result (if it holds up, which it won't) imply necessarily breaking of LI? If the limiting speed is that of the superluminal neutrinos, it seems SR still applies, except the photon is slightly massive. This probably has even more severe problems that breaking LI, but still...
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user566Sep 24 '11 at 4:59

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I don't think it's a stupid question. I discussed it with several people in person today who probably wouldn't want me posting their names and off-the-cuff thoughts on the internet. But one obstacle is that SN 1987A tells us that (lower-energy) neutrinos and photons go at about the same speed in empty space, so if you wanted to invoke something that slightly slows down photons, you need it to slow down neutrinos in almost exactly the same way except, for some reason, in the context of this experiment. Seems like a stretch, but I don't know if anyone's thought about it very systematically.
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Matt ReeceSep 24 '11 at 5:03

@Slaviks: Speaking for myself, this is a good description. The only thing perhaps not widely enough appreciated is the enormity of this problem: the large number of relevant operators and the ridiculous precision in which they are bounded makes “unlikely” quite the understatement. Which is great - constraints on ultra-high energy physics are hard to come by, and LI proves to be a very useful such constraint.
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user566Oct 6 '11 at 20:21

Luboš Motl and Matt Reece have given excellent answers to this question. I just wanted to add an answer about a relatively recent proposal.

Stephen Gubser wrote a very recent prerint in the style of a 'no-go theorem' that discusses the major issues with superluminal travel in extra-dimensions; Link. His main point is that it is "hard" to find a compact spacetime manifold that satisfies the null-energy condition (in extra-dimensions) and satisfies Einstein's equations. However, there are a few loopholes that Gubser points out:

A spacetime dimension of $D=6$ may allow superluminal travel

One likely cannot have a smooth manifold as a model of spacetime; we need singularities. This likely rules out other 'common' possibilities for spacetime such as the conifold (since it resolves to a $CY_3$ as per the famous construction by Candelas and de la Ossa)

Abstract
Motivated by a recent neutrino OPERA experimental finding, which showed that neutrino particles travel faster than light, I here propose a modified Newtonian model, which adopts a relativistic perspective. Assuming that there is no absolute frame of reference, I use Newton’s laws of motion to derive a novel result for relative time. The derived result yields a precise prediction of the velocity of neutrino reported in the aforementioned neutrino OPERA experiment.

I took a look at your paper and it looks like a nice manipulation of equations (i did not check for consistency). Which has been done quite a lot recently to describe the Opera results. This however does not answer the question above, if there are any Neutrino Models describing this effect.
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MichaelJan 31 '12 at 12:43